The Veronese Surface in PG(5,3) and Witt's 5-$(12,6,1$ Design

TL;DR

This paper constructs a point model for Witt's 5-(12,6,1) design using the Veronese surface in PG(5,3), providing a coordinate-free approach and a projective representation of the Mathieu group M12.

Contribution

It introduces a novel coordinate-free construction of Witt's design and a projective representation of M12 based on the Veronese surface in PG(5,3).

Findings

Constructs a point model for Witt's design using the Veronese surface.

Provides a projective representation of the Mathieu group M12.

Offers an easy, coordinate-free approach to known results.

Abstract

A conic of the Veronese surface in PG(5,3) is a quadrangle. If one such quadrangle is replaced with its diagonal triangle, then one obtains a point model $K$ for Witt's 5-$(12,6,1)$ design, the blocks being the hyperplane sections containing more than three (actually six) points of $K$. As such a point model is projectively unique, the present construction yields an easy coordinate-free approach to some results obtained independently by H.S.M. Coxeter and G. Pellegrino, including a projective representation of the Mathieu group $M_{12}$ in PG(5,3).